On a heat conduction problem by Myshkis

Let tau(1) = 1, tau(2), tau(3), ... be recurrently defined by Sigma(j=1)(n)(Sigma(s=j)(n)tau s)(-1/2) = 1 for n = 2, 3, ... The sequence {tau(n)} appeared in a heat conduction problem studied by Myshkis (J. Differ. Equations Appl. 3, 1997, 89-91), He proposed as an open problem the asymptotic expression for tau(n). It is shown in this paper that tau(n) is increasing and lim(n) tau(n)/n = pi(2)/2. AMS Subject Classification: 39A10, 35K05, 93B52.